Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 804642, 15 pages
doi:10.1155/2012/804642
Research Article
A Hybrid Extragradient-Like Method for
Variational Inequalities, Equilibrium
Problems, and an Infinitely Family of Strictly
Pseudocontractive Mappings
Yaqin Wang,1, 2 Hongkun Xu,3 and Xiaoli Fang1
1
Department of Mathematics, Shaoxing University, Shaoxing 312000, China
Mathematical College, Sichuan University, Sichuan, Chengdu 610064, China
3
Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan
2
Correspondence should be addressed to Yaqin Wang, wangyaqin0579@126.com
Received 31 December 2011; Accepted 15 January 2012
Academic Editor: Yonghong Yao
Copyright q 2012 Yaqin Wang et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The purpose of this paper is to consider a new scheme by the hybrid extragradient-like method
for finding a common element of the set of solutions of a generalized mixed equilibrium problem,
the set of solutions of a variational inequality, and the set of fixed points of an infinitely family
of strictly pseudocontractive mappings in Hilbert spaces. Then, we obtain a strong convergence
theorem of the iterative sequence generated by the proposed iterative algorithm. Our results
extend and improve the results of Issara Inchan 2010 and many others.
1. Introduction
Let H be a real Hilbert space with inner product ·, · and norm · . Let C be a nonempty
closed convex subset of H and A be a mapping of C into H. We denote by FA the set of
fixed points of A and by PC the metric projection of H onto C. We also denote by R the set of
all real numbers.
Recall the following definitions.
i A is called monotone if
Ax − Ay, x − y ≥ 0,
∀x, y ∈ C.
1.1
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ii A is called α-inverse-strongly monotone if there exists a positive constant α such
that
2
Ax − Ay, x − y ≥ αAx − Ay ,
∀x, y ∈ C.
1.2
iii A is called k-Lipschitz continuous if there exists a positive constant k such that
Ax − Ay ≤ kx − y,
∀x, y ∈ C.
1.3
Clearly, every inverse strongly monotone mapping is Lipschitz continuous and monotone.
A mapping T : C → C is said to be ξ-strictly pseudocontractive if there exists a
constant ξ ∈ 0, 1 such that
T x − T y2 ≤ x − y2 ξI − T x − I − T y2 ,
∀x, y ∈ C.
1.4
It is known that if T is a 0-strictly pseudocontractive mapping, then T is a nonexpansive mapping. So the class of ξ-strictly pseudocontractive mappings includes the class of
nonexpansive mappings.
Let Θ : C × C → R be a bifunction. The equilibrium problem for Θ is to find that x ∈ C
such that
Θ x, y ≥ 0,
∀y ∈ C.
1.5
The set of solutions of problem 1.5 is denoted by EP.
Given a mapping A : C → H, let Θx, y Ax, y − x for all x, y ∈ C. Then problem
1.5 reduces to the following classical variational inequality problem of finding x ∈ C such
that
Ax, y − x ≥ 0,
∀y ∈ C.
1.6
The set of solutions of problem 1.6 is denoted by VIC, A.
Numerous problems in physics, optimization, saddle point problems, complementarity problems, mechanics, and economics reduce to find a solution of problem 1.5. Many
methods have been proposed to solve problem 1.5; see, for instant, 1–3. In 1997, Combettes
and Hirstoaga 4 introduced an iterative scheme of finding the best approximation to initial
data when EP is nonempty and proved a strong convergence theorem.
Recently, Peng and Yao 5 introduced the following generalized mixed equilibrium
problem of finding x ∈ C such that
Θ x, y ϕ y − ϕx Bx, y − x ≥ 0,
∀y ∈ C,
1.7
where B : C → H is a nonlinear mapping, and ϕ : C → R is a function. The set of solutions
of problem 1.7 is denoted by G MEP.
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In the case of B 0 and ϕ 0, then problem 1.7 reduces to problem 1.5. In the case
of Θ 0, ϕ 0, and B A, then problem 1.7 reduces to problem 1.6. In the case of ϕ 0,
problem 1.7 reduces to the generalized equilibrium problem. In the case of B 0, problem
1.7 reduces to the following mixed equilibrium problem of finding x ∈ C such that
Θ x, y ϕ y − ϕx ≥ 0,
∀y ∈ C,
1.8
which was considered by Ceng and Yao 6. The set of sulutions of this problem is denoted
by MEP.
The problem 1.7 is very general in the sense that it includes, as special cases, some
optimization, variational inequalities, minimax problems, the Nash equilibrium problem in
noncooperative games, economics, and others see, for instant, 6
Recently, S. Takahashi and W. Takahashi 7 introduced the following iteration process:
x1 u ∈ C,
1
Θ un , y Axn , y − un y − un , un − xn ≥ 0, ∀y ∈ C,
r
xn1 βn xn 1 − βn Sαn u 1 − αn un , n ≥ 1,
1.9
and used this iteration process to find a common element of the set of fixed points of a
nonexpansive mapping S and the set of solutions of a generalized equilibrium problem in
a Hilbert space.
In 2008, Bnouhachem et al. 8 introduced the following new extragradient iterative
method. Let C be a closed convex subset of a real Hilbert H, A be an α-inverse strongly
monotone mapping of C into H, and let S be a nonexpansive mapping of C into itself such
that FS ∩ VIC, A /
∅. Let the sequences {xn }, {yn } be given by
x1 , u ∈ C chosen arbitrary,
xn1
yn PC xn − λn Axn ,
βn xn 1 − βn S αn u 1 − αn PC xn − λn Ayn ,
1.10
∀n ≥ 1,
where {αn }, {βn }, and {λn } ⊂ 0, 1 satisfy some parameters controlling conditions. They
proved that the sequence {xn } converges strongly to a common element of FS ∩ VIC, A.
In 2010, Ceng et al. 9 introduced the following hybrid extragradient-like method. Let
C be a nonempty closed convex subset of a real Hilbert space H, A : C → H be a monotone,
k-Lipschitz continuous mapping, and let S : C → C be a nonexpansive mapping such that
FS ∩ VIC, A / ∅. Let the sequences {xn }, {yn }, and {zn } be defined by
x0 ∈ C,
yn 1 − γn xn γn PC xn − λn Axn ,
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zn 1 − αn − βn xn αn yn βn SPC xn − λn Ayn ,
Cn z ∈ C : zn − z2 ≤ xn − z2 3 − 3γn αn b2 Axn 2 ,
Qn {z ∈ C, xn − z, x0 − xn ≥ 0},
xn1 PCn ∩Qn x0 ,
∀n ≥ 0.
1.11
Under the suitable conditions, they proved the sequences {xn }, {yn }, {zn } converge strongly
to the same point PFS∩VIC,A x0 .
In 2010, Inchan 10 introduced a new iterative scheme by the hybrid extragradient
method in a Hilbert space H as follows: x0 ∈ H, C1 C ⊂ H, x1 PC x0 , and let
un ∈ C,
1
Θ un , y Bxn , y − un y − un , un − xn ≥ 0,
rn
∀y ∈ C,
yn PC un − λn Aun ,
zn αn xn 1 − αn S βn xn 1 − βn PC un − λn Ayn ,
1.12
Cn1 {z ∈ Cn : zn − z ≤ xn − z},
xn1 PCn1 x0 ,
∀n ≥ 0,
where {αn }, {βn }, and {λn } ⊂ 0, 1 satisfy some parameters controlling conditions. They
proved that {xn } and {un } strongly converge to the same common element of the set of fixed
points of a nonexpansive mapping, the set of solutions of an equilibrium problem, and the
set of solutions of the variational inequality for nonexpansive mappings.
Very recently, Wang 11 defined the mapping Wn as follows:
Un,n1 I,
Un,n γn Tn Un,n1 1 − γn I,
Un,n 1 − γn−1 I,
Un,n−1 γn−1 Tn−1
..
.
Un,k γk Tk Un,k1 1 − γk I,
Un,k 1 − γk−1 I,
Un,k−1 γk−1 Tk−1
..
.
Un,2 γ2 T2 Un,3 1 − γ2 I,
Wn Un,1 γ1 T1 Un,2 1 − γ1 I,
1.13
where γ1 , γ2 , . . . are real numbers such that 0 ≤ γn ≤ 1, Ti θi I 1 − θi Ti , where Ti is a μi strictly pseudocontractive mapping of C into itself and θi ∈ μi , 1. It follows from 12 that
Ti is nonexpansive and FTi FTi . Nonexpansivity of each Ti ensures the nonexpansivity
of Wn .
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Motivated and inspired by the above work, in this paper, we introduced the following
new iterative scheme by the extragradient-like method: C1 C ⊂ H, x1 PC x0 ,
1
un ∈ C such that Θ un , y Bxn , y − un ϕ y − ϕun y − un , un − xn ≥ 0,
rn
yn 1 − sn un sn PC un − λn Aun ,
zn αn xn 1 − αn Wn βn xn 1 − βn PC un − λn Ayn ,
Cn1 z ∈ Cn : zn − z2 ≤ xn − z2 31 − sn b2 Aun 2 ,
xn1 PCn1 x0 ,
y ∈ C,
∀n ≥ 0,
1.14
where A, B : C → H and A is a monotone, k-Lipschitz continuous mapping, and B is a
β-inverse strongly monotone mapping. Then under the suitable conditions, we derive some
strong convergence results.
2. Preliminaries
Let C be a nonempty closed and convex subset of a Hilbert space H, for any x ∈ H, and there
exists a unique nearest point in C, denoted by PC x such that
x − PC x ≤ x − y,
∀y ∈ C.
2.1
The projection operator PC : H → C is nonexpansive. Moreover, PC x is characterized by the
following properties: for every x ∈ H and y ∈ C,
x − y2 ≥ x − PC x2 y − PC x2 ,
x − PC x, y − PC x ≤ 0.
2.2
2.3
Suppose that A is monotone and continuous. Then the solutions of the variational
inequality VIC, A can be characterized as solutions of the so-called Minty variational
inequality:
x∗ ∈ VIC, A ⇐⇒ Ax, x − x∗ ≥ 0,
∀x ∈ C.
2.4
In what follows, we shall make use of the following lemmas.
Lemma 2.1. Let H be a real Hilbert space. Then for any x, y ∈ H, we have
i x ± y2 x2 ± 2x, y y2 ,
ii tx 1 − ty2 tx2 1 − ty2 − t1 − tx − y2 , for all t ∈ 0, 1.
We denote by NC v the normal cone for C at a point v ∈ C, that is NC v : {x∗ ∈ E∗ :
v − y, x∗ ≥ 0 for all y ∈ C}. In the following, we shall use the following Lemma.
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Lemma 2.2 see 13. Let C be a nonempty closed convex subset of a Banach space E, and let A be a
monotone and hemicontinuous operator of C into E∗ . Let T ⊂ E × E∗ be an operator defined as follows:
Av NC v,
Tv ∅,
v ∈ C,
v∈
/ C.
2.5
Then T is maximal monotone, and T −1 0 VIC, A.
Lemma 2.3 see 14. Let C be a nonempty closed convex subset of a strictly convex Banach space
E. Let T1 , T2 , . . . be nonexpansive mappings of C into itself such that ∞
/ ∅ and γ1 , γ2 , . . . be
i1 FTi real numbers such that 0 < γi ≤ b < 1 for every i 1, 2, . . .. Then for any x ∈ C and k ∈ N, the limit
limn → ∞ Un,k exists.
Using Lemma 2.3, define the mapping W of C into itself as follows:
Wx : lim Wn x lim Un,1 x,
n→∞
n→∞
x ∈ C.
2.6
Such a mapping W is called the modified W mapping generalized by T1 , T2 , . . . , γ1 , γ2 , . . ., and
θ1 , θ2 , . . ..
Lemma 2.4 see 14. Let C be a nonempty closed convex subset of a strictly convex Banach space
∅ and γ1 , γ2 , . . . be
E. Let T1 , T2 , . . . be nonexpansive mappings of C into itself such that ∞
i1 FTi /
real numbers such that 0 < γi ≤ b < 1 for every i 1, 2, . . .. Then W is a nonexpansive mapping
satisfying that FW ∞
i1 FTi .
Lemma 2.5 see 15. Let C be a nonempty closed convex subset of a strictly convex Banach space
∅ and γ1 , γ2 , . . . be
E. Let T1 , T2 , . . . be nonexpansive mappings of C into itself such that ∞
i1 FTi /
real numbers such that 0 < γi ≤ b < 1 for every i 1, 2, . . .. If K is any bounded subset of C, then
lim supWx − Wn x 0.
n → ∞ x∈K
2.7
For solving the equilibrium problem, let us assume that Θ satisfies the following conditions:
H1 Θx, x 0 for all x ∈ C,
H2 Θ is monotone, that is, Θx, y Θy, x ≤ 0 for all x, y ∈ C,
H3 for each y ∈ C, x → Θx, y is weakly upper semicontinuous,
H4 for each x ∈ C, y → Θx, y is convex and lower semicontinuous,
A1 for each x ∈ H and r > 0, there exists a bounded subset Dx ⊂ C and yx ∈ C such that for
any z ∈ C \ Dx ,
1
Θ z, yx ϕ yx − ϕz yx − z, z − x < 0,
r
A2 C is a bounded set.
2.8
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Lemma 2.6 see 6. Let C be a closed subset of H. Let ϕ : C → R be a lower semicontinuous and
convex function, and Θ be a bifunction from C × C to R satisfying (H1)–(H4). For r > 0 and x ∈ H,
define a mapping Tr : H → C as follows:
Tr x 1
z ∈ C : Θ z, y ϕ y − ϕz y − z, z − x ≥ 0, ∀y ∈ C
r
2.9
for all x ∈ H. Assume that either (A1) or (A2) holds. Then the following results hold:
∅ for each x ∈ H, and Tr is single valued,
1 Tr x /
2 Tr is firmly nonexpansive, that is, for all x, y ∈ H, Tr x − Tr y2 ≤ Tr x − Tr y, x − y,
3 FTr MEP,
4 MEP is closed and convex.
3. Strong Convergence Theorems
Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a
bifunction from C × C into R satisfying (H1)–(H4) and ϕ : C → R be a lower semicontinuous
and convex function with (A1) or (A2). Let A : C → H be a monotone, k-Lipschitz continuous
mapping and B : C → H be a β-inverse-strongly monotone mapping. Let Ti : C → C be a μi ∅ and {γi } be a real
strictly pseudocontractive mapping with F : ∞
i1 FTi ∩ VIC, A ∩ G MEP /
sequence such that 0 < γi ≤ b < 1, for all i ≥ 1. Assume that the control sequences {αn }, {βn },
{sn } ⊂ 0, 1, {rn } ⊂ 0, 2β, and {λn } ⊂ 0, 1/2k satisfy the following conditions:
i lim supn → ∞ αn < 1, lim supn → ∞ βn < 1,
ii 0 < a ≤ λn ≤ b < 1/2k, 0 < d ≤ rn ≤ e < 2β,
iii sn → 1n → ∞ and sn > 3/4 for all n ≥ 0.
Then the sequence {xn } defined by 1.14 converges strongly to PF x0 .
Proof. We divide the proof into several steps.
Step 1 {xn } is well defined. Indeed, for any q ∈ F. Put tn PC un − λn Ayn . Since un Trn xn − rn Bxn , q Trn q − rn Bq, and B is β-inverse-strongly monotone and rn ∈ 0, 2β, for
any n ≥ 0, we have
un − q2 Tr xn − rn Bxn − Tr q − rn Bq 2
n
n
2
≤ xn − rn Bxn − q − rn Bq 2
2
≤ xn − q − 2rn Bxn − Bq, xn − q rn2 Bxn − Bq
2
2
≤ xn − q rn rn − 2β Bxn − Bq
2
≤ xn − q .
3.1
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It follows from 2.2 and 2.4 that
2 2
tn − q2 ≤ un − λn Ayn − q − un − λn Ayn − tn 2
un − q − un − tn 2 − 2λn Ayn , tn − q
2 2 2
un − q − un − yn − yn − tn − 2λn Ayn , yn − q
− 2 un − yn , yn − tn 2λn Ayn , yn − tn
2 2 2
≤ un − q − un − yn − yn − tn 2 un − yn − λn Ayn , tn − yn .
3.2
In addition, we have
un − yn − λn Ayn , tn − yn un − yn − λn Aun , tn − yn λn Aun − Ayn , tn − yn
≤ sn un − λn Aun − PC un − λn Aun , tn − yn
λn sn − 1 Aun , tn − yn λn kun − yn tn − yn ,
3.3
and by 2.3, we obtain
un − λn Aun − PC un − λn Aun , tn − yn
un − λn Aun − PC un − λn Aun , 1 − sn tn − un sn tn − PC un − λn Aun ≤ 1 − sn λn Aun tn − yn yn − un 3.4
sn un − λn Aun − PC un − λn Aun , tn − PC un − λn Aun ≤ 1 − sn λn Aun tn − yn yn − un .
It follows from 3.2–3.4 that
tn − q2 ≤ un − q2 − un − yn 2 − yn − tn 2 2sn 1 − sn λn Aun tn − yn yn − un 2λn 1 − sn Aun tn − yn 2λn kun − yn tn − yn 2 2 2
≤ un − q − un − yn − yn − tn 2 2 sn 1 − sn 2b2 Aun 2 tn − yn yn − un 1 − sn b2 Aun 2 tn − yn 2 kb tn − yn 2 yn − un 2
un − q2 − sn − bkun − yn 2 − 2sn − 1 − kbtn − yn 2 31 − sn b2 Aun 2
≤ un − q2 31 − sn b2 Aun 2 .
3.5
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Setting wn βn xn 1 − βn tn . Therefore, from 1.14, 3.1, and 3.5, we get the following:
zn − q2 ≤ αn xn − q2 1 − αn Wn wn − q2
2
2
≤ αn xn − q 1 − αn wn − q
2
2
2
≤ αn xn − q 1 − αn βn xn − q 1 − αn 1 − βn tn − q
2
≤ αn 1−αn βn 1−αn 1−βn xn − q 31 − αn 1 − βn 1 − sn b2 Aun 2
2
≤ xn − q 31 − sn b2 Aun 2 .
3.6
So, q ∈ Cn and hence F ⊂ Cn for all n ≥ 1. It is easy to see that Cn is closed and convex for all
n ≥ 1. This implies that {xn } and {un } are well defined.
Step 2 {xn } is a Cauchy sequence. It is easy to see that F is closed and convex. From xn1 PCn1 x0 ∈ Cn1 ⊂ Cn and xn PCn x0 , for any q ∈ F, we have
3.7
xn − x0 ≤ xn1 − x0 ≤ q − x0 .
So {xn } is bounded, and limn → ∞ xn − x0 exists. So it follows from 3.1, 3.6, and the
continuity of A that {un }, {zn }, and {Aun } are bounded. By the construction of Cn , we have
Cm ⊂ Cn and xm PCm x0 ∈ Cn for any positive integer m ≥ n. So from 2.2, we have
xm − xn 2 ≤ xm − x0 2 − xn − x0 2 .
3.8
Letting m, n → ∞ in 3.8, we have xm − xn → 0, which implies that {xn } is a Cauchy
sequence. So there exists z ∈ C such that xn → zn → ∞.
Step 3 limn → ∞ wn − Wwn 0. From 3.8, we have
lim xn − xn1 0.
3.9
n→∞
Since xn1 ∈ Cn1 , by 3.9 and condition iii, we obtain that
zn − xn1 ≤ xn − xn1 31 − sn b2 Aun 2 −→ 0
n −→ ∞.
3.10
So
lim zn − xn 0.
3.11
zn − xn 1 − αn xn − Wn wn ,
3.12
n→∞
Since
from 3.11 and condition i, we have
lim xn − Wn wn 0.
n→∞
3.13
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For any q ∈ F, from 3.1 and 3.5, we obtain that
wn − q2 ≤ βn xn − q2 1 − βn tn − q2
2 2
≤ βn xn − q 1 − βn un − q 31 − sn b2 Aun 2
2 2
≤ xn − q − 1 − βn rn 2β − rn Bxn − Bq 31 − sn b2 Aun 2 .
3.14
Therefore, we have
zn − q2 ≤ αn xn − q2 1 − αn wn − q2
3.15
2
2
≤ xn − q − 1 − αn 1 − βn rn 2β − rn Bxn − Bq 31 − sn b2 Aun 2 ,
which implies that
2
1 − αn 1 − βn rn 2β − rn Bxn − Bq
2 2
≤ xn − q − zn − q 31 − sn b2 Aun 2
≤ zn − xn zn − q xn − q 31 − sn b2 Aun 2 .
3.16
Combining the above inequality, 3.11 and conditions i–iii, we have
lim Bxn − Bq 0.
n→∞
3.17
It follows from Lemma 2.6 that
un − q2 Tr xn − rn Bxn − Tr q − rn Bq 2
n
n
≤ xn − rn Bxn − q − rn Bq , un − q
1 xn −rn Bxn − q−rn Bq 2 un −q2 − xn −rn Bxn − q − rn Bq − un − q 2
2
1 xn − q2 un − q2 − xn − un 2 2rn xn − un , Bxn − Bq − rn2 Bxn − Bq2 .
≤
2
3.18
Therefore,
un − q2 ≤ xn − q2 − xn − un 2 2rn xn − un , Bxn − Bq − rn2 Bxn − Bq2 .
3.19
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11
By 3.5 and 3.19, we have
zn − q2 ≤ αn xn − q2 1 − αn wn − q2
2
2 2 ≤ αn xn − q 1 − αn βn xn − q 1 − βn tn − q
2
2 2
≤ αn xn − q 1 − αn βn xn − q 1 − βn un − q 31 − sn b2 Aun 2
2
≤ xn − q − 1 − αn 1 − βn xn − un 2 2rn 1 − αn 1 − βn xn − un , Bxn − Bq
2
− 1 − αn 1 − βn rn2 Bxn − Bq 31 − sn b2 Aun 2 .
3.20
It follows from 3.20 that
2 2
1 − αn 1 − βn xn − un 2 ≤ xn − q − zn − q 2rn 1 − αn 1 − βn xn − un , Bxn − Bq
2
− 1 − αn 1 − βn rn2 Bxn − Bq 31 − sn b2 Aun 2
≤ zn − xn zn − q xn − q 2exn − un Bxn − Bq
2
e2 Bxn − Bq 31 − sn b2 Aun 2 .
3.21
Therefore, from 3.11, 3.17, 3.21, and conditions i, iii,
lim xn − un 0,
n→∞
3.22
which implies that un → zn → ∞. And from 3.1, 3.5, and 3.6, we have
zn − q2 ≤ αn xn − q2 1 − αn βn xn − q2 1 − αn 1 − βn tn − q2
2
2
≤ xn − q − 1 − αn 1 − βn sn − bkun − yn 2
− 1 − αn 1 − βn 2sn − 1 − kbtn − yn 31 − sn b2 Aun 2 .
3.23
Thus it follows that
2
2
1 − αn 1 − βn sn − bkun − yn 1 − αn 1 − βn 2sn − 1 − kbtn − yn 2 2
≤ xn − q − zn − q 31 − sn b2 Aun 2
≤ zn − xn zn − q xn − q 31 − sn b2 Aun 2 .
3.24
Therefore, from 3.11, 3.24 and conditions i–iii, we obtain limn → ∞ un − yn 0 and
limn → ∞ tn − yn 0. Furthermore, we have
lim tn − un 0,
n→∞
3.25
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which implies that tn → zn → ∞. It follows from 3.22 and 3.25 that
lim tn − xn 0.
3.26
n→∞
Note that wn − xn 1 − βn tn − xn , so by 3.26 and condition i, we obtain that
lim wn − xn 0,
3.27
n→∞
which implies that wn → zn → ∞. Note that
wn − Wwn ≤ wn − xn xn − Wn wn Wn wn − Wwn .
3.28
Therefore, by 3.13, 3.27, 3.28 and Lemma 2.5, we have
lim wn − Wwn 0.
3.29
n→∞
Step 4 z ∈ F. Since wn → zn → ∞ and W is nonexpansive, by 3.29, we have
z − Wz ≤ z − wn wn − Wwn Wwn − Wz −→ 0 n −→ ∞.
3.30
So z Wz, that is, z ∈ FW ∞
i1 FTi .
Next we show that z ∈ G MEP. Indeed, from H2 and 1.14, we get
1
y − un , un − xn ≥ Θ y, un .
Bxn , y − un ϕ y − ϕun rn
3.31
Put yt ty 1 − tz, t ∈ 0, 1 and y ∈ C. So yt ∈ C. By 3.31, we have
Byt , yt − un
un − xn
≥ Byt , yt − un − Bxn , yt − un −ϕ yt ϕun − yt − un ,
Θ yt , un
rn
Byt − Bun , yt − un Bun − Bxn , yt − un
un − xn
Θ yt , un
− ϕ yt ϕun − yt − un ,
rn
un − xn Θ yt , un .
≥ −Bun − Bxn yt − un − ϕ yt ϕun − yt − un r
n
3.32
Let n → ∞ in 3.32, since B is nonexpansive and ϕ is lower semicontinuous, by 3.22,
condition ii and H4, we have
Byt , yt − z ≥ −ϕ yt ϕz Θ yt , z .
3.33
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13
So, from H1, H4, and the above inequality, we obtain
0 Θ yt , yt ϕ yt − ϕ yt
≤ tΘ yt , y 1 − tΘ yt , z tϕ y 1 − tϕz − ϕ yt
t Θ yt , y ϕ y − ϕ yt 1 − t Θ yt , z ϕz − ϕ yt
≤ t Θ yt , y ϕ y − ϕ yt 1 − tt Byt , y − z ,
3.34
Θ yt , y ϕ y − ϕ yt 1 − t Byt , y − z ≥ 0.
3.35
that is,
Letting t → 0 in the above inequality, we obtain for each y ∈ C,
Θ z, y ϕ y − ϕz Bz, y − z ≥ 0.
3.36
This implies that z ∈ G MEP.
Finally, we show that z ∈ VIC, A. Define a mapping T as Lemma 2.2. Let v, u ∈
GT . Since u − Av ∈ NC v and tn ∈ C, we have v − tn , u − Av ≥ 0. Since tn PC un − λn Ayn ,
we have
v − tn , tn − un − λn Ayn ≥ 0,
3.37
and hence
v − tn , u ≥ v − tn , Av
t n − un
≥ v − tn , Av − v − tn ,
Ayn
λn
t n − un
v − tn , Av − Ayn −
λn
t n − un
v − tn , Av − Atn v − tn , Atn − Ayn − v − tn ,
λn
t n − un .
≥ −v − tn Atn − Ayn − v − tn λ
n
3.38
Since limn → ∞ tn − yn 0, and A is Lipschitz continuous, by 3.25 and condition ii, we
deduce that v − z, u ≥ 0. Since T is maximal monotone, we have z ∈ T −1 0 and so z ∈
VIC, A. Hence z ∈ F.
Step 5 z PF x0 . Put z∗ PF x0 . Since xn PCn x0 , z ∈ F and the norm is lower semicontinuous, we have
z∗ − x0 ≤ z − x0 ≤ lim infxn − x0 lim xn − x0 ≤ z∗ − x0 ,
n→∞
n→∞
3.39
14
Journal of Applied Mathematics
that is, z∗ − x0 z − x0 . Hence z z∗ PF x0 , since z∗ is the unique element in F that
minimizes the distance from x0 .
Thus, {xn } converges strongly to PF x0 .
Remark 3.2. Theorem 3.1 mainly improves the results of Inchan 10. To be more precise,
Theorem 3.1 improves and extends Theorem 3.1 of 10 from the following several aspects:
i from a single nonexpansive mapping to an infinite family of strictly pseudocontractive mappings,
ii from generalized equilibrium problems to generalized mixed equilibrium problems,
iii from hybrid extragradient methods to hybrid extragradient-like methods,
iv the condition of A relaxes to monotone, Lipschitz continuous.
4. Application
Theorem 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a
bifunction from C × C into R satisfying (H1)–(H4) and ϕ : C → R be a lower semicontinuous
and convex function with (A1) or (A2). Let A : C → H be a monotone, k-Lipschitz continuous
mapping and T : C → C be a ξ-strictly pseudocontractive mapping. Let Ti : C → C be a μi -strictly
∅ and {γi } be a real sequence
pseudocontractive mapping with F : ∞
i1 FTi ∩ VIC, A ∩ G MEP /
such that 0 < γi ≤ b < 1, for all i ≥ 1. Let the sequence {xn } be generated C1 C ⊂ H, x1 PC x0 ,
un ∈ C such that Θ un , y I − T xn , y − un
1
ϕ y − ϕun y − un , un − xn ≥ 0, y ∈ C,
rn
yn PC un − λn Aun ,
zn αn xn 1 − αn Wn βn xn 1 − βn PC un − λn Ayn ,
4.1
Cn1 {z ∈ Cn : zn − z ≤ xn − z},
xn1 PCn1 x0 ,
∀n ≥ 0.
Assume that the control sequence {αn }, {βn } ⊂ 0, 1, {rn } ⊂ 0, 1 − ξ and {λn } ⊂ 0, 1/2k satisfy
the following conditions:
i lim supn → ∞ αn < 1, lim supn → ∞ βn < 1,
ii 0 < a ≤ λn ≤ b < 1/2k, 0 < d ≤ rn ≤ e < 1 − ξ.
Then {xn } converges strongly to PF x0 .
Proof. A ξ-strictly pseudocontractive mapping is 1 − ξ/2-inverse-strongly monotone. Then
taking sn 1, for all n ≥ 1 in Theorem 3.1, we obtain the conclusion.
Acknowledgment
The authors are extremely grateful to the referees for their useful suggestions that improved
the content of the paper.
Journal of Applied Mathematics
15
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